TOPICS
Search

Catenoid


catenoid

A catenary of revolution. The catenoid and plane are the only surfaces of revolution which are also minimal surfaces. The catenoid can be given by the parametric equations

x=ccosh(v/c)cosu
(1)
y=ccosh(v/c)sinu
(2)
z=v,
(3)

where u in [0,2pi).

The line element is

 ds^2=cosh^2(v/c)dv^2+c^2cosh^2(v/c)du^2.
(4)

The first fundamental form has coefficients

E=c^2cosh^2(v/c)
(5)
F=0
(6)
G=cosh^2(v/c),
(7)

and the second fundamental form has coefficients

e=-c
(8)
f=0
(9)
g=1/c.
(10)

The principal curvatures are

kappa_1=1/csech^2(v/c)
(11)
kappa_2=-1/csech^2(v/c).
(12)

The mean curvature of the catenoid is

 H=0
(13)

and the Gaussian curvature is

 K=-1/(c^2)sech^4(v/c).
(14)
HelicoidCatenoid

The helicoid can be continuously deformed into a catenoid with c=1 by the transformation

x(u,v)=cosalphasinhvsinu+sinalphacoshvcosu
(15)
y(u,v)=-cosalphasinhvcosu+sinalphacoshvsinu
(16)
z(u,v)=ucosalpha+vsinalpha,
(17)

where alpha=0 corresponds to a helicoid and alpha=pi/2 to a catenoid.

This deformation is illustrated on the cover of issue 2, volume 2 of The Mathematica Journal.


See also

Catenary, Costa Minimal Surface, Helicoid, Minimal Surface, Surface of Revolution

Explore with Wolfram|Alpha

References

do Carmo, M. P. "The Catenoid." §3.5A in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 43, 1986.Fischer, G. (Ed.). Plate 90 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, p. 86, 1986.Geometry Center. "The Catenoid." http://www.geom.umn.edu/zoo/diffgeom/surfspace/catenoid/.GRAPE. "Catenoid." http://www-sfb256.iam.uni-bonn.de/grape/EXAMPLES/AMANDUS/catenoid.html.GRAPE. "Catenoid-Helicoid Deformation." http://www-sfb256.iam.uni-bonn.de/grape/EXAMPLES/AMANDUS/cathel.html.Gray, A. "The Catenoid." §20.4 Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 467-469, 1997.JavaView. "Classic Surfaces from Differential Geometry: Catenoid/Helicoid." http://www-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_CatenoidHelicoid.html.Meusnier, J. B. "Mémoire sur la courbure des surfaces." Mém. des savans étrangers 10 (lu 1776), 477-510, 1785.Ogawa, A. "Helicatenoid." Mathematica J. 2, 21, 1992.Osserman, R. A Survey of Minimal Surfaces. New York: Dover, p. 18 1986.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 247-249, 1999.

Cite this as:

Weisstein, Eric W. "Catenoid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Catenoid.html

Subject classifications